AMBIGUITY FUNCTION AND FRAME THEORETIC PROPERTIES OFPERIODIC ZERO AUTOCORRELATION WAVEFORMS

JOHN J. BENEDETTO AND JEFFREY J. DONATELLINORBERT WIENER CENTER

DEPARTMENT OF MATHEMATICSUNIVERSITY OF MARYLAND, COLLEGE PARK 20742

JJB@MATH.UMD.EDU

Abstract. Constant Amplitude Zero Autocorrelation (CAZAC) waveforms u are analyzed in

terms of the ambiguity function Au. Elementary number theoretic considerations illustrate that

peaks in Au are not stable under small pertubations in its domain. Further, it is proved that the

analysis of vector-valued CAZAC waveforms depends on methods from the theory of frames. Fi-

nally, techniques are introduced to characterize the structure of Au, to compute u in terms of Au,

and to evaluate MSE for CAZAC waveforms.

1. Introduction

1.1. Background. Let N ≥ 1 be an integer, let ZN be the additive group Z of integers modulo

N , and let F be the field R of real numbers or the field C of complex numbers. (ZN is also a

commutative ring with unit; and it is a field if and only if N is prime.) Let d ≥ 1 be an integer.

We shall construct and analyze N -periodic functions u : ZN → Fd, which are unimodular and have

0-autocorrelation for each m ∈ ZN \ {0}. The autocorrelation Au of u is defined by

∀ m ∈ ZN , Au(m) =1N

N−1∑

k=0

< u(m + k), u(k) >,

where each u(k) = (u1(k), . . . , ud(k)), where uj(k) ∈ F, k ∈ ZN , and j = 1, . . . , d, and where the

inner product is

< u(k), u(m) > =d∑

j=1

uj(k)uj(m).

Thus, the norm of each u(k) is ||u(k)|| =< u(k), u(k) >1/2, and we say that u is unimodular when

||u|| =1.

If u : ZN → Fd is unimodular and has 0-autocorrelation for each m ∈ ZN \ {0}, then u is

a CAZAC (Constant Amplitude Zero Autocorrelation) waveform in Fd of length N . This is a

generalization of the usual setting of CAZAC waveforms or codes in which F = C and d = 1. In

this latter case, CAZAC waveforms (and some of their close relatives) are also called by the following1

names among others: polyphase codes with good periodic or optimum correlation properties, e.g.,

[42], [39], [26], [16], [9]; perfect autocorrelation or root-of-unity sequences, e.g., [33], [17], [21]; bi-

unimodular sequences, e.g., [4], [5], [6], [24]; bent functions, e.g., [11], [10]. The literature in this

area is extensive, one might say overwhelming. A hint of its breadth and activity in the study of

CAZAC waveforms is found in [27]. Fundamental applications are to radar and communications

theory, e.g., [31] and [34], [30], respectively.

In this paper we prove new results for two classes of CAZAC waveforms, which describe the

behavior of their ambiguity functions. These are called the Wiener [45] and Milewski classes

[32]. We also give first results on our problem of describing vector-valued CAZAC waveforms

u : ZN → Fd, which are also finite unit norm tight frames (FUNTFs). FUNTFs are a basic

model for applications dealing with robust transmission of data over erasure channels such as the

internet [22], [7], multiple antenna code design for wireless communications [28], multiple description

coding [43], [23], and quantum detection and information [12], [14], [13]. Many of the results invoke

elementary number theory for their verification.

In Subsection 1.2 we give requisite definitions and in Subsection 1.3 we describe our results. Our

overall goal is to provide several new and useful techniques in waveform design.

We were led to this topic because of three related aspects of our work. These are: waveform

design in radar [29]; Σ−∆ quantization for FUNTFs [3] and the potential theoretic characterization

of FUNTFs [2]; and our perspective of Norbert Wiener’s Generalized Harmonic Analysis [45] vis a

vis recent, deep work characterizing CAZAC waveforms by Bjorck and Saffari [4], [5], [6], [35], [37]

and Haagerup [24], [25].

Remark 1.1. a. Our emphasis is on F = R or F = C. On the other hand, it is natural to do what

we have done for finite extensions of the p-adic rationals, Qp, or for the finite fields of p integers

under modular addition, where p is prime. It is also natural to conduct our analysis on locally

compact, and, hence, complete fields F. The completeness is essential since every d-dimensional

Hilbert space H over F is isometric to Fd; and so it becomes interesting to investigate our results

directly for functions u : ZN → H, when the intrinsic properties of H are required as opposed to

results valid up to isometries.

b. As we have indicated, this paper deals with periodic waveforms. Aperiodic waveforms or codes

with properties similar to periodic CAZAC waveforms are especially important in many modern

applications. An aperiodic waveform u : Z → C is one that is compactly supported. The sequel

to the present paper deals with the aperiodic case, focusing on the themes developed herein, but2

in the context of Golay-Shapiro-Welti codes [19], [20], [41], and [44], which date from 1949, 1951,

and 1960, respectively. These closely related and justly famous codes have been reinvented and

renamed (alas) in recent years, see [38] for a documentation of this history sometimes repeating

itself.

1.2. Definitions. Gauss was able to construct CAZAC waveforms of any length N , see [6], [36].

In fact, if we let

M =

N, N odd,

2N, N even,

then it is elementary to prove that

(1.1) ∀k ∈ ZN , u(k) = e2πik2/M

defines a CAZAC waveform of length N , see Theorem 3.1. Wiener [45] used such examples as a

staple in his theory of Generalized Harmonic Analysis. Because of his deep analysis of autocorre-

lation in this context, we refer to (1.1) as a Wiener CAZAC waveform . When N is odd, a slightly

more general Wiener CAZAC waveform, than (1.1), is defined by

∀k ∈ ZN , u(k) = e2πi(ak2+bk)/N ,

where a, b ∈ Z and (a,N) = 1, i.e., a and N are relatively prime.

Further, we can use any CAZAC waveform, {v(k)}M−1k=0 , to generate a family of CAZAC wave-

forms of length MN2, N = 2, 3, . . .. To be precise, let v be a CAZAC waveform of length M , and

define the waveform

(1.2)

∀k ∈ ZMN2 , u(k) = u(aN +b) = v(a)e2πiab/(MN), k = aN +b, a = 0, . . . ,MN−1, b = 0, . . . , N−1.

This is equivalent to

∀k ∈ ZMN2 , u(k) = v

(⌊k

N

⌋mod M

)e2πib k

N c(k mod N)/(MN),

where bxc is the greatest integer nx ≤ x. Milewski [32] verified that u is a CAZAC waveform and,

as such, we refer to waveforms u defined by (1.2) as Milewski CAZAC waveforms of length MN2,

see [1] for user-friendly software generating and analyzing Milewski CAZAC waveforms.

A natural generalization of autocorrelation is the ambiguity function, so useful in radar. We

shall define it with the following normalization.

3

Definition 1.2. Let u : ZN → Cd be an N -periodic waveform. The ambiguity function Au :

ZN × ZN → C of u is defined by

∀(m,n) ∈ ZN × ZN , Au(m,n) =1N

N−1∑

k=0

< u(m + k), u(k) > e2πikn/N .

Note that the autocorrelation Au(m) is Au(m, 0). Notationally, we shall use j, k,m as “time”

variables, and n, q, r as “frequency” variables. Also, δ(j) = 0 if j = 0 and δ(0) = 1. Further,

a 6 | b indicates that the integer a does not divide the integer b; and 1S designates the characteristic

function of the set S.

CAZAC waveforms can be thought of in terms of FUNTFs as we do in Section 5. There is an

even more basic relation in terms of tight frames due to the characterization of CAZAC waveforms

as circulant Hadamard matrices with complex entries (which can be thought of as tight frames),

see [6] for the characterization. As such, we now define finite frames.

Definition 1.3. a. A sequence {x(j)}Nj=1 ⊆ Fd is a finite frame for Fd if {x(j)}N

j=1 spans Fd.

b. Let {x(j)}Nj=1 ⊆ Fd. The Bessel (analysis) operator, L : Fd → `2(ZN ), for {x(j)}N

j=1 is defined

by

∀x ∈ Fd, L(x) = {< x, x(j) >}Nj=1,

where `2(ZN ), the space of F-valued functions on ZN , can be identified with F× . . .×F (N -times).

L is the N × d matrix operator L = (x(1) . . . x(N))τ , where τ designates the (complex) adjoint. The

adjoint (synthesis) operator, L∗ : `2(ZN ) → Fd, of L is characterized by the equation,

∀v = {v(j)}Nj=1 ∈ `2(ZN ), L∗(v) =

N∑

j=1

v(j)x(j) ∈ Fd,

where v is considered as an N × 1 vector. L∗ is the d×N matrix operator L∗ = (x(1), . . . , x(N)) ∈Fd × . . .× Fd (N -times).

c. Let {x(j)}Nj=1 ⊆ Fd. S = L∗L : Fd → Fd is the frame operator and G = LL∗ : `2(ZN ) → `2(ZN )

is the Gram operator for {x(j)}Nj=1.

It is straightforward to calculate that

∀x ∈ Fd, S(x) =N∑

j=1

< x, x(j) > x(j) ∈ Fd

and

∀v ∈ `2(ZN ), G(v) =N∑

j=1

< x(k), x(j) >v(j) ∈ `2(ZN ).

4

G =(< x(k), x(j) >

)N

k,j=1is the N ×N Gram matrix operator .

The following result is well-known, e.g., [2], [8].

Proposition 1.4. Let {x(j)}Nj=1 ⊆ Fd.

a. {x(j)}Nj=1 is a frame for Fd if and only if

∃ A,B > 0 such that ∀x ∈ Fd, A||x||2 ≤N∑

j=1

| < x, x(j) > |2 ≤ B||x||2.

b. {x(j)}Nj=1 is a frame for Fd if and only if S is a bijection on Fd.

c. If {x(j)}Nj=1 is a frame for Fd, then G : L(Fd) → L(Fd) is a bijection, and

∀x ∈ Fd, x =N∑

j=1

< x, S−1x(j) > x(j) =N∑

j=1

< x, x(j) > S−1x(j) = L∗G−1(L(x)).

A and B in Proposition 1.4a are frame constants . If {x(j)}Nj=1 is a frame for Fd and if A = B

in Proposition 1.4a, then {x(j)}Nj=1 is a tight frame. In this case, the decomposition in Proposition

1.4c can be simplified to

∀x ∈ Fd, x =d

N

N∑

j=1

< x, x(j) > x(j).

We shall also use the Discrete Fourier Transform (DFT).

Definition 1.5. The N ×N DFT matrix DN is

DN =(e−2πimn/N

)N−1

m,n=0.

The DFT v of v ∈ `2(ZN ) is defined by

∀ n ∈ ZN , v(n) =N−1∑

m=0

v(m)e−2πimn/N ,

i.e., v = DNv.

1.3. Results. The results of Section 2 are well known. They are included for perspective and the

fact that our point of view of dealing with the nonabelian group GN may be new. Specifically, we

want to emphasize that there are many non-chirp-like CAZAC waveforms.

Theorem 3.3 and its consequences, Corollary 3.4 and Example 3.5, as well as the accompanying

Figures 1–4 for Wiener waveforms, are new. The impact of these results is that even chirp-like

CAZAC waveforms are not stable under small pertubations in the domain ZN × ZN of the ambi-

guity function. Theorems 4.1, 4.2, and 4.3, as well as the accompanying Figures 5–8, provide the5

analogous more compliacted behavior of Milewski waveforms.

In Section 5 we relate CAZAC waveforms with the theory of frames in a fundamental problem

we have posed with an eye to vector-valued and multidimensional waveform design (and emerging

applications). Theorems 5.3, 5.4, and 5.5 provide partial solutions to this problem.

Section 6 introduces several applicable techniques to characterize the structue of the ambiguity

function matrix (Propositions 6.1, 6.2, 6.3, 6.4, 6.6 and Example 6.8). We also determine the signal

u for given Au data (Theorem 6.5 and Remark 6.7), and compute mean square error (MSE) for

ZAC waveforms (Theorem 6.9), see Proposition 2.4.

2. CAZAC waveforms in C

Consider mappings α : `2(ZN ) → `2(ZN ) of the following form:

a. Rotation ρa(u) = au by a ∈ C, |a| = 1;

b. Translation (cyclic shift) τ−j(u) of u ∈ `2(ZN ) by j ∈ Z, defined as τ−j(u)(k) = u(k + j);

c. Decimation (permutation subgroup) πj(u) of u ∈ `2(ZN ) by j ∈ Z, (j,N) = 1, defined as

πj(u)(k) = u(jk);

d. Linear frequency modulation µq,ω(u) of u ∈ `2(ZN ) by q ∈ Z and any Nth root of unity ω,

defined as µq,ω(u)(k) = ωkqu(k);

e. Conjugation κ(u) = u.

Clearly, we have

(2.1) Aρa(u)(m) = Au(m),

(2.2) Aτ−j(u)(m) = Au(m),

(2.3) Aπj(u)(m) = Au(jm), (j,N) = 1,

(2.4) Aµq,ω(u)(m) = ωmqAu(m),

(2.5) Aκ(u)(m) = Au(m) = Au(−m) = Au(N −m),

where jm and N −m are evaluated mod N . Equation (2.3) is a consequence of the fact that ZN

is identified with {kj : 0 ≤ k ≤ N − 1} in the case (j, N) = 1.

Theorem 2.1 is an elementary extension of (2.1) – (2.5) to the ambiguity function.

6

Theorem 2.1. Let u : `2(ZN ) → `2(ZN ) and let m,n ∈ ZN .

a. If |a| = 1, then Aρa(u)(m,n) = Au(m,n).

b. If j ∈ Z, then Aτ−j(u)(m,n) = e−2πijn/NAu(m,n).

c. Let 1 ≤ j ≤ N − 1. If (j, N) = 1, then the multiplicative inverse j−1 ∈ ZN exists and

Aπj(u)(m,n) = Au(jm, n/j), where jm and n/j are evaluated mod N .

d. If q ∈ Z and ω is an Nth root of unity, then Aµq,ω(u)(m,n) = ωmqAu(m,n).

e. Aκ(u)(m,n) = Au(m,−n) = e−2πimn/NAu(−m,n) = e−2πimn/NAu(N −m,n).

Definition 2.2. Let u, v ∈ `2(ZN ). u and v are equivalent if v can be obtained from u by a finite

composition (finite succession) α of the mappings ρa, τ−j , πj , µq,ω, and κ. The set of all such finite

compositions α is denoted by GN .

This notion of equivalence defines an equivalence relation R ⊆ `2(ZN )× `2(ZN ) on `2(ZN ); and

GN is a nonabelian group which is the focus of a sequel by the authors.

Remark 2.3. Define the partition, {Xρ,u : u ∈ `2(ZN )}, by the rule that Xρ,u = {v ∈ `2(ZN ) :

∃ |a| = 1 such that v = au}. The assertion that {Xρ,u} is a partition means that the sets Xρ,u,

u ∈ `2(ZN ), are disjoint and their union over u is `2(ZN ). {Xρ,u} defines an equivalence relation

Rρ ⊆ `2(ZN )× `2(ZN ) whose equivalence classes Xρ,u satisfy

Xρ,u ⊆ XR,u = {v : ∃ α ∈ GN such that α(u) = v},

where {XR,u} is the partition of `2(ZN ) defined by the equivalence relation R.

There is the following compelling problem for Rρ, and therefore associated with the notion of

equivalence in Definition 2.2: For a given N , compute or estimate the number of nonequivalent

CAZAC waveforms. The problem has been investigated by Gabidulin [17], [18]. Bjorck and Saffari

[6] proved that if N = MK2 then there are infinitely many nonequivalent CAZAC waveforms, e.g.,

N = 8, 9, or 12. (The case N = 4 is straightforward). On the other hand, Haagerup [25] has given

a complete mathematical proof that if N is prime, then there are only finitely many nonequivalent

CAZAC waveforms.

The following two propositions are well known and elementary to verify.

Proposition 2.4. If {u(k)}N−1k=0 ⊆ C has constant amplitude (CA) on ZN , e.g., if u is unimodular,

then its DFT has the zero autocorrelation property (ZAC), i.e., Au(m) = δ(m) for m ∈ ZN . If

{u(k)}N−1k=0 has the zero autocorrelation property (ZAC), then its DFT has constant amplitude

(CA). Thus, {u(k)}N−1k=0 is a CAZAC waveform if and only if its DFT u is a CAZAC waveform.

7

Proposition 2.5. Let (N1, N2) = 1 and let u : ZN1 → C and v : ZN2 → C be CAZAC waveforms.

Then w = uv : ZN → C, N = N1N2 is a CAZAC waveform.

This method of making new longer CAZAC waveforms from given CAZAC waveforms is not the

same as the Milewski method.

Example 2.6. a. Let N be odd. Binary CAZAC waveforms u : ZN → {±1} can not exist. In

fact, if u is constant, then Au = 1 on ZN . If u is not constant, then∑N−1

k=0 u(m + k)u(k) is a sum

with N terms, each taking the value ±1; as such Au(m) 6= 0 for m ∈ ZN \ {0}. It is easy to see

that Au(m) 6= 0 for m ∈ ZN \ {0}. Even more, it is elementary to verify that Au(m) is odd for

m ∈ ZN \ {0}.b. Let N be arbitrary. It is well known that if u : ZN → {±1}, then Au(m) = N mod 4. Thus,

if Au(m) has zeros, and, in particular, if u is a CAZAC waveform, then 4 divides N .

c. Let N be arbitrary. The only known binary CAZAC waveform u : ZN → {±1}, up to any

translation and multiplication by −1, is {1, 1, 1,−1}. Also, it is clear that any Milewski waveform

generated by {1, 1, 1,−1} is not binary. However, there do exist periodic complex binary sequences,

not ±1, which are CAZAC waveforms, e.g., Bjorck (1985) [4], [5] and Golomb (1992) [21], cf., [24].

In fact, Saffari [35], [37] was able to find all such complex binary sequences.

3. Ambiguity function analysis of Wiener CAZACs

As indicated in Subsection 1.2, the Wiener waveforms defined in (1.1) have long been known

to be CAZAC waveforms. This assertion can be also stated in terms of primitive roots of unity,

which we do in Theorem 3.1; and the proof is elementary, and, hence, omitted. Since this extension

of (1.1) in terms of primitive roots is so natural, we shall also refer to such functions as Wiener

waveforms.

Theorem 3.1. Given N ≥ 1. Let

M =

N, N odd,

2N, N even,

and let ω be a primitive Mth root of unity. Define the Wiener waveform u : ZN → C by u(k) = ωk2,

0 ≤ k ≤ N − 1. Then u is a CAZAC waveform.

Remark 3.2. Periodic waveforms similar to Wiener waveforms have long been used in engineering,

e.g., in continuous wave (CW) radar (see Chapter 10 of [31]). Such waveforms go back to Frank8

(1953) with signifigant subsequent contributions through the 1970s by Heimiller (1961) [26], Frank

and Zadoff (1962) [16], Schroeder (1970) [40], Chu (1972) [9], and Frank (1973) [15].

Theorem 3.3. Let j ∈ Z. Define u : ZN × ZN → C by uj(k) = e2πijk2/M , where M = 2N if N is

even and M = N if N is odd. If N is even, then

Auj (m,n) =

e2πijm2/(2N), jm + n ≡ 0 mod N,

0, otherwise.

If N is odd

Auj (m, n) =

e2πijm2/N , 2jm + n ≡ 0 mod N,

0, otherwise.

Proof. Let N be even, and set uj(k) = eπijk2/N . We calculate

Auj (m,n) =1N

N−1∑

k=0

uj(m + k)uj(k)e2πikn/N

=1N

N−1∑

k=0

e(πi/N)(jm2+2jkm+2kn) = eπijm2/N 1N

N−1∑

k=0

e2πik(jm+n)/N .

If jm + n ≡ 0 mod N , then1N

N−1∑

k=0

e2πik(jm+n)/N = 1.

Otherwise, we have1N

N−1∑

k=0

e2πik(jm+n)/N =e(2πi(jm+n)/N)N − 1e2πi(jm+n)/N − 1

= 0.

Let N be odd, and set u(k) = e2πik2/N . We calculate

Auj (m,n) =1N

N−1∑

k=0

uj(m + k)uj(k)e2πikn/N

=1N

N−1∑

k=0

e(2πi/N)(jm2+2jkm+kn) = e2πijm2/N 1N

N−1∑

k=0

e2πik(2jm+n)/N .

If 2jm + n ≡ 0 mod N , then1N

N−1∑

k=0

e2πik(2jm+n)/N = 1.

9

Otherwise, we have1N

N−1∑

k=0

e2πik(2jm+n)/N =e2πi(2m+n)/N)N − 1e(2πi(2m+n)/N − 1

= 0.

¤

The ambiguity function, Au, of a Wiener CAZAC waveform u : ZN → C, as given in Theorem

3.1, has a simple behavior since, for any fixed value of n, Au(m,n) is zero for all except one value

of m. That is, for each fixed n, the graph of Au(•, n) as a function of m consists of a single peak,

see Figures 1 and 2. In fact, we have the following consequence of Theorem 3.3.

Corollary 3.4. Let {u(k)}N−1k=0 be a Wiener CAZAC waveform as given in Theorem 3.1. (In

particular, ω is a primitive M -th root of unity.)

If N is even, then

Au(m, n) =

ωm2, m ≡ −n mod N,

0, otherwise.

If N is odd, then

Au(m,n) =

ωm2, m ≡ −n(N + 1)/2 mod N,

0, otherwise.

Example 3.5. a. Let N be odd and let ω = e2πi/N . Then, u(k) = ωk2, 0 ≤ k ≤ N − 1, is a

CAZAC waveform. By Corollary 3.4, |Au(m,n)| = |ωm2 | = 1 if 2m + n = lm,nN for some lm,n ∈ Zand |Au(m,n)| = 0 otherwise, i.e., Au(m, n) = 0 on ZN × ZN unless 2m + n ≡ 0 mod N . In the

case 2m + n = lm,nN for some lm,n ∈ Z, we have the following phenomenon. If 0 ≤ m ≤ N−12 and

2m + n = lm,nN for some lm,n ∈ Z, then n is odd; and if N+12 ≤ m ≤ N − 1 and 2m + n = lm,nN

for some lm,n ∈ Z, then n is even. Thus, the values (m,n) in the domain of the ambiguity function

Au, for which Au(m,n) = 0, appear as two parallel discrete lines. The line whose domain is

0 ≤ m ≤ N−12 has odd function values n; and the line whose domain is N+1

2 ≤ m ≤ N − 1 has even

function values n, see Figure 3, cf., Figure 4.

b. The behavior observed in part a has extensions for primitive and nonprimitive roots of unity.

Let u : ZN → C be a Wiener waveform. Thus, u(k) = ωk2, 0 ≤ k ≤ N − 1, and ω = e2πij/M ,

(j, M) = 1, where M is defined in terms of N in Theorem 3.1. By Corollary 3.4, for each fixed

n ∈ ZN , the function Au(•, n) of m vanishes everywhere except for a unique value mn ∈ ZN for

which |Au(mn, n)| = 1.

The hypotheses of Theorem 3.3 do not assume that e2πij/M is a primitive Mth root of unity.

In fact, in the case that e2πij/M is not primitive, then, for certain values of n, Au(•, n) will be10

identically 0 and, for certain values of n, |Au(•, n)| = 1 will have several solutions, see Figure 5,

cases j = 2, 4, 25, 98. For example, if N = 100 and j = 2, then, for each odd n, Au(•, n) = 0 as a

function of m. If N = 100 and j = 3, then (100, 3) = 1 so that e2πi3/100 is a primitive 100th root

of unity; and, in this case, for each n ∈ ZN there is a unique mn ∈ ZN such that |Au(mn, n)| = 1

and Au(m,n) = 0 for each m 6= mn.

4. Ambiguity function analysis of Milewski CAZACs

The following theorem shows that for a fixed value of n, the ambiguity function, Au(m,n), of a

Milewski CAZAC waveform is well behaved in the sense that it vanishes for a large set of values of

m, see Figure 6.

Theorem 4.1. Let {u(k)}MN2

k=1 be a Milewski CAZAC waveform generated by a CAZAC waveform

v : ZM → C. If m,n ∈ Z have the property that N 6 | (m + n), then Au(m,n) = 0.

Proof. We first write Au as follows:

Au(m,n) =1

MN2

MN2−1∑

k=0

u(m + k)u(k)e2πikn/(MN2)

=1

MN2

MN2−1∑

k=0

v

(⌊m + k

N

⌋mod M

)v

(⌊k

N

⌋mod M

)eη,

where η = 2πiMN (

⌊m+k

N

⌋(m+k mod N)− ⌊

kN

⌋(k mod N)+ kl

N ). Next write each k as k = γ + sMN ,

where γ = 0, 1, . . . , MN − 1, and s = 0, 1, . . . , N . Then

Au(m,n) =1

MN2

MN−1∑

γ=0

N−1∑

s=0

v(⌊

m + γ + sMN

N

⌋mod M)v(

⌊γ + sMN

N

⌋mod M)eη

=1

MN2

MN−1∑

γ=0

N−1∑

s=0

v(⌊

m + γ

N

⌋mod M)v(

⌊ γ

N

⌋mod M)eη,

where

η =2πi

MN

(⌊m + γ + sMN

N

⌋(m + γ + sMN mod N)−

⌊γ + sMN

N

⌋(γ + sMN mod N) +

(γ + sMN)nN

)

=2πi

MN

((⌊m + γ

N

⌋+ sM

)(m + γ mod N)−

(⌊ γ

N

⌋+ sM

)(γ mod N) =

γn

N+ sMn

)

=2πi

MN

(⌊m + γ

N

⌋(m + γ mod N)−

⌊ γ

N

⌋(γ mod N) +

γn

N+ sM (m + γ mod N − γ mod N + n)

),

11

β(m,n, γ) = v

(⌊m + γ

N

⌋mod M

)v

(⌊ γ

N

⌋mod M

)eδ(m,n,γ),

and

δ(m,n, γ) =2πi

MN

(⌊m + γ

N

⌋(m + γ mod N)−

⌊ γ

N

⌋(γ mod N) +

γn

N

).

Therefore,

Au(m,n) =1

MN2

MN−1∑

γ=0

β(m,n, γ)N−1∑

s=0

(e

2πiN

((m+γ) mod N−γ mod N+n)))s

.

If N 6 | (m + γ mod N − γ mod N + n), i.e., N 6 | (m + n), then

Au(m,n) =1

MN2

MN−1∑

γ=0

β(m,n, γ)

(e

2πiN

((m+γ)mod N−γ mod N+n))N

− 1

e2πiN

((m+γ)mod N−γmod N+n) − 1= 0.

¤

The graphs of the ambiguity function of Milewski CAZAC waveforms are periodic. The precise

nature of this periodicity is the content of the following result.

Theorem 4.2. Let {u(k)}MN2

k=1 be a Milewski CAZAC waveform generated by a CAZAC waveform

v : ZM → C, and let s ∈ Z. Then

∀(m,n) ∈ ZN × ZN , |Au(m,n)| = |Au(m− sMN, n + sMN)|.

Proof. By definition,

Au(m− sMN, n + sMN) =1

MN2

MN2−1∑

k=0

u (m− sMN + k) u(k)e2πi

MN2 k(n+sMN)

=1

MN2

MN2−1∑

k=0

v

(⌊m + k

N

⌋mod M

)v

(⌊k

N

⌋)eη′ ,

where

η′ =2πi

MN

(⌊m− sMN + k

N

⌋(m− sMN + k mod N)−

⌊k

N

⌋(k mod N) +

k(n + sMN)N

)

=2πi

MN

((⌊m + k

N

⌋− sM

)(m + k mod N)−

⌊k

N

⌋(k mod N) +

kn)N

+ sMk

)

=2πi

MN

(⌊m + k

N

⌋(m + k mod N)−

⌊k

N

⌋(k mod N) +

kn

N

)+

2πi

MN(−sM(m + k mod N) + sMk) .

12

We can write η′ as

η′ = η +2πi

N(k − (m + k mod N))

= η +2πi

N(k −m− r(k)N) = η − 2πism

N+ 2πisr(k)

for some integer r(k) which depends on k, and where

η =2πi

MN

(⌊m + γ + sMN

N

⌋(m + γ + sMNmodN)−

⌊γ + sMN

N

⌋(γ + sMNmodN) +

(γ + sMN)nN

).

Therefore,

eη′ = eηe2πism/Ne2πir(k) = eηe−2πism/N .

Now,

|Au(m− sMN,n + sMN)| =∣∣∣∣∣∣

1MN2

MN2−1∑

k=0

v

(⌊m + k

N

⌋mod M

)v

(⌊k

N

⌋mod M

)eηe

−2πismN

∣∣∣∣∣∣

= |e−2πismN Au(m,n)| = |Au(m,n)|.

¤

Let K = MN2. We would like to be able to say that

(4.1) ∀m,n ∈ Z, MN 6 | (m + n) ⇒ Au(m,n) = 0

and

(4.2) ∀m,n, s ∈ Z, |Au(m,n)| = |Au(m− sN, n + sN)|.

The reason (4.1) and (4.2) are attractive is that they say the following. The quantity |Au(m,n)| is

N -periodic as a function of n, i.e., there are at most N different graphs of |Au(m,n)|. Also, for a

fixed n, the values of m in which |Au(m,n)| may be non-zero are MN periodic, i.e., the non-zero

points of the graph of |Au(m,n)| are separated by MN values of m. (4.1) and (4.2) do hold for a

large number of CAZAC waveforms, see Figure 7. However, this is not always the case, see Figure

8. In fact, we have the following result.

Theorem 4.3. Let {u(k)}MN2−1k=0 be a Milewski CAZAC waveform generated by an M -periodic

Wiener CAZAC waveform. Then (4.1) and (4.2) are valid if M is even.13

Proof. Part (4.1). Let M be even. Assume M divides m + n. Then, by the proof of Theorem 4.1,

Au(m) =1

MN

MN−1∑γ=0

eτ ,

where

τ =πi

M

[(⌊m + γ

N

⌋mod M

)2

−(⌊ γ

N

⌋mod M

)2

+2N

⌊m + γ

N

⌋(m + γ mod N)− 2

N

⌊ γ

N

⌋(γ mod N) +

2γn

N2

].

Write each γ = ν + rN , where ν = 0, . . . , N − 1 and r = 0, . . . , M − 1. We compute (modulo 2πi)

τ =πi

M

[(⌊m + ν

N

⌋+ r mod M

)2

−(⌊ ν

N

⌋+ r mod M

)2

+2N

(⌊m + ν

N

⌋+ r

)(m + ν mod N)

− 2N

(⌊ ν

N

⌋+ r

)(ν mod N) +

2νn

N2+

2nr

N

]

=πi

M

[(⌊m + ν

N

⌋+ r − c(r)M

)2

− r2 +2N

(⌊m + ν

N

⌋+ r

)(m + ν mod N)− 2rν

N+

2νn

N2+

2nr

N

]

=πi

M

[(⌊m + ν

N

⌋+ r

)2

− 2(⌊

m + ν

N

⌋+ r

)c(r)M + c2(r)M2 − r2 +

2N

(⌊m + ν

N

⌋+ r

)(m + ν mod N)

−2rν

N+

2νn

N2+

2nr

N

]

=πi

M

[⌊m + ν

N

⌋2

+ 2r

⌊m + ν

N

⌋+

2N

(⌊m + ν

N

⌋+ r

)(m + ν mod N)− 2rν

N+

2νn

N2+

2nr

N

]

=2πir

MN

[N

⌊m + ν

N

⌋+ (m + ν mod N)− ν + n

]+ α(m,n, ν)

=2πi

M

((m + n)r

N

)+ α(m,n, ν),

where α(m,n, ν) is a constant independent of r and c(r) whose value depends on r. If M 6 | (m + n), then

Au(m,n) =1

MN

N−1∑ν=0

eα(m,n,ν)M−1∑r=0

e2πi m+nN r/M =

1MN

N−1∑ν=0

eα(m,n,ν) e(2πi m+n

N /M)M − 1e2πi(m+n)/M − 1

= 0.

Part (4.2). If MN 6 | (m + n) then we are done by (4.1) since m− sN + l + sN = m + l. Assume

MN | (m + n). Then, by the proof of part (4.1), we have

Au(m,n) =1N

N−1∑

ν=0

eλ,

where

λ =πi

M

[⌊m + ν

N

⌋2

+2N

⌊m + ν

N

⌋(m + ν mod N) +

2νn

N2

].

We calculate

|Au(m− sN, n + sN)| =∣∣∣∣∣1N

N−1∑

ν=0

eλ′∣∣∣∣∣ ,

14

where

λ′ =πi

M

[⌊m− sN + ν

N

⌋2

+2N

⌊m− sN + ν

N

⌋(m− sN + ν mod N) +

2νn + sN

N2

]

=πi

M

[⌊m + ν

N

⌋2

+2N

⌊m + ν

N

⌋(m + ν mod N) +

2νn

N2− 2s

⌊m + ν

N

⌋+ s2 − 2s

N(m + ν mod N) +

2νs

N

]

= λ +πi

M

[s2 − 2s

N

(N

⌊m + ν

N

⌋+ (m + ν mod N)

)+

2νs

N

]

= λ +πi

M

[s2 − 2s(m + ν)

N+

2νs

N

]= λ +

πi

M

[s2 − 2sm

N

].

Therefore, we have that

|Au(m− sN, n + sN)| =∣∣∣∣∣e

πiM

(s2− 2smN

) 1N

N−1∑

ν=0

eλ

∣∣∣∣∣ = |Au(m,n)|.

¤

5. Vector valued CAZAC waveforms and frames

Our results in this section rely on the following two well known facts, e.g., see [2], [8].

Proposition 5.1. Let {x(k)}Nk=1 ⊆ Fd, with corresponding Bessel operator L : Fd → `2(ZN ),

x 7→ {< x, x(k) >}Nk=1. Then {x(k)}N

k=1 ⊆ Fd is a tight frame for Fd with frame constant C if and

only if L∗L = CId, where Id is the d× d identity matrix.

Proposition 5.2. Let {x(k)}Nk=1 ⊆ Fd be a FUNTF for Fd, then its frame constant is N

d .

It is desirable to generate tight frames which also have the properties of CAZAC waveforms. We

are able to generate such sequences in Cd, which, in fact, provide a new class of FUNTFs.

Theorem 5.3. Let u = {u(k)}Nk=1 be a CAZAC waveform in C. Define

∀ k = 1, . . . , N, v(k) = v(k) =1√d

(u(k) u(k + 1) . . . u(k + d− 1)) .

Then v = {v(k)}Nk=1 ⊆ Cd is a CAZAC waveform in Cd and {v(k)}N

k=1 is a FUNTF for Cd with

frame constant Nd .

Proof. (CAZAC) Clearly,

||v(j)|| =

k+d−1∑

k=j

|u(k)/√

d|2

1/2

=

j+d−1∑

k=j

1/d

1/2

= 1.

15

Consider Av(m), where m 6≡ 0 mod N .

Av(m) =1N

N−1∑

k=0

< v(m+k), v(k) > =1N

N−1∑

k=0

d−1∑

h=0

u(m + h + k)u(h + k)

=1N

d−1∑

h=0

N−1∑

k=0

u(m + h + k)u(h + k) =d−1∑

h=0

Au(m) = 0.

(FUNTF) Consider the operator L∗L, defined by the Bessel operator L : Cd → CN , x 7→ {<x, x(k) >}N

k=1. Using the corresponding matrix operator (Definition 1.3b) and the definition of v(k),

k = 1, . . . , N , we have the following calculation, which depends on the periodicity of u:

L∗L =1d

u(1) u(2) . . . u(N)

u(2) u(3) . . . u(N + 1)...

.... . .

...

u(d) u(d + 1) . . . u(N + d− 1)

u(1) u(2) . . . u(d)

u(2) u(3) . . . u(d + 1)...

.... . .

...

u(N) u(N + 1) . . . u(N + d− 1)

=1d

NAu(0) NAu(1) . . . NAu(d− 1)

NAu(−1) NAu(0) . . . NAu(d− 2)...

.... . .

...

NAu(−d + 1) NAu(−d + 2) . . . NAu(0)

=N

dId.

Thus, we invoke Proposition 5.2 to complete the proof. ¤

The following result provides a method for generating nontrivial vector valued CAZAC wave-

forms. By nontrivial, we mean that the elements in each row do not form a CAZAC waveform

themselves. This could also allow for the possibility of discovering different families of CAZAC

waveforms in C if the appropriate tight frame and vector valued CAZAC waveform is chosen. This

is a consequence of Theorem 5.5.

Theorem 5.4. Let {x(k)}Nk=1 ⊆ Cd be a FUNTF for Cd, with frame constant C and with associated

Bessel map L : Cd → `2(ZN ); and let u = {u(j)}Mj=1 ⊆ Cd be a CAZAC waveform in Cd (u : ZM →

Cd). Then L(u(j)) ∈ `2(ZN ), j = 1, . . . , M , and { 1√C

L(u(j))}Mj=1 ⊆ CN (= `2(ZN )) is a CAZAC

waveform in CN .

Proof. Let Lj = L(u(j)) ∈ CN , j = 1, . . . , M , and let Lj = (Lj1, Lj2, . . . , LjN ), so that Ljk =<

u(j), x(k) >.

(CA) Using the hypothesis that {x(k)}Nk=1 is a FUNTF, we calculate the `2(ZN )-norm of each

16

L(u(j)), j = 1, . . . , M :

||L(u(j))||2 = ||Lj ||2 =N∑

k=1

|Ljk|2 =N∑

k=1

| < u(j), x(k) > |2 = C||u(j)||2 = C.

(ZAC) Also, we compute

A 1√C

L◦u(m) =1

CM

M−1∑

j=0

< L(u(m + j)), L(u(j)) >

=1

CM

M−1∑

j=0

< u(m + j), L∗L(u(j)) >=C

CM

M−1∑

j=0

< u(m + j), u(j) >

= Au(m) = δ(m),

where L∗L = CId by Proposition 5.1 and where the last step follows by the CAZAC hypothesis on

u. ¤

Theorem 5.5. Let {x(k)}Nk=1 ⊆ Cd be a FUNTF for Cd, with frame constant C and with associated

Bessel map L : Cd → `2(ZN ); and let u = {u(j)}Mj=1 ⊆ CN be a CAZAC waveform in CN for which

{u(j)}Mj=1 ⊆ L(Cd). Then L∗(u(j)) ∈ `2(Zd), j = 1, . . . , M , and { 1√

CL∗ ◦ u(j)}M

j=1 ⊆ Cd(= `2(Zd))

is a CAZAC waveform in Cd.

Proof. For each j = 1, . . . ,M , let u(j) = (u(j)1, . . . , u(j)N ). Since u(j) ∈ Range (L), there is

yj ∈ Cd such that L(yj) = u(j) ∈ CN and each u(j)k =< yj , x(k) >, k = 1, . . . , N . Therefore, for

each j = 1, . . . , M ,

||L∗(u(j))||2 = ||N∑

k=1

< yj , x(k) > x(k)||2 = ||Cyj ||2 = CN∑

k=1

| < yj , x(k) > |2 = C||u(j)||2 = C,

where we have used the tight frame hypothesis once and the unit norm condition once.

Since L∗L = CId on Cd (Proposition 5.1), we obtain LL∗ = CIN on L(Cd) ⊆ CN by taking

c ∈ L(Cd), choosing yc ∈ Cd for which L(yc) = c, and computing LL∗(c) = L(Cxc) = Cc.

We use LL∗ = CIN on L(Cd) in the following calculation.

A 1√C

L∗◦u(m) =1

CM

M−1∑

j=0

< L∗(u(m + j)), L∗(u(j)) >

=1

CM

M−1∑

j=0

< LL∗(u(m + j)), u(j) >

=1M

M−1∑

j=0

< u(m + j), u(j) >= δ(m).

17

¤

6. Statistic, Inversion of Au, and RMS error

Propositions 6.1, 6.2, and 6.3 can be used to devise statistics to analyze Doppler [29].

Proposition 6.1. Let u = {u(k)}N−1k=0 ⊆ C be a CAZAC waveform. Then

∀n ∈ Z,

N−1∑

m=0

|Au(m, n)|2 = 1.

Proof. Fix n ∈ Z. We compute

N−1∑

m=0

|Au(m,n)|2 =N−1∑

m=0

(1N

N−1∑

k=0

u(m + k)u(k)e2πikn/N

) 1

N

N−1∑

j=0

u(m + j)u(j)e2πijn/N

=1

N2

N−1∑

m=0

N−1∑

k=0

N−1∑

j=0

u(m + k)u(k)u(m + j)u(j)e2πin(k−j)/N

=1

N2

N−1∑

k=0

N−1∑

j=0

u(j)u(k)e2πin(k−j)/NN−1∑

m=0

u(m + k)u(m + j)

=1

N2

N−1∑

k=0

N−1∑

j=0

u(j)u(k)e2πin(k−j)/NNAu(k − j)

=1

N2

N−1∑

k=0

N−1∑

j=0

u(j)u(k)e2πin(k−j)/NNδ(k − j)

=1N

N−1∑

k=0

u(k)u(k) = 1.

¤

Similarly, but more simply, we can verify the following result, which does not require u to be a

CAZAC waveform.

Proposition 6.2. Let u = {u(k)}N−1k=0 ⊆ C. Then

∀m ∈ Z,N−1∑

n=0

|Au(m,n)|2 = 1.

Proposition 6.3. Let {u(k)}N−1k=0 ⊆ C. Then

a.

∀n ∈ ZN ,N−1∑

m=0

Au(m,n) =1N

(N−1∑

k=0

u(k)

)u(n)

18

b.

∀m ∈ ZN ,N−1∑

n=0

Au(m,n) = u(m)u(0).

Proof. a.N−1∑

m=0

Au(m,n) =1N

N−1∑

m=0

N−1∑

k=0

u(m + k)u(k)e2πikn/N

=1N

N−1∑

k=0

u(k)e2πikn/NN−1∑

m=0

u(m + k) =1N

N−1∑

k=0

u(k)e2πikn/NN−1∑

m=0

u(m) =1N

(N−1∑

m=0

u(m)

)u(n).

b.N−1∑

n=0

Au(m,n) =1N

N−1∑

n=0

N−1∑

k=0

u(m + k)u(k)e2πikn/N

=1N

N−1∑

k=0

u(m + k)u(k)N−1∑

n=0

e2πikn/N =1N

N−1∑

k=0

u(m + k)u(k)Nδ(k) = u(m)u(0).

¤

The following result is a consequence of Proposition 6.3.

Proposition 6.4. Let u = {u(k)}N−1k=0 ⊆ C.

a. u is a ZAC waveform if and only if

∀n ∈ ZN ,

∣∣∣∣∣N−1∑

m=0

Au(m,n)

∣∣∣∣∣ = 1

b. u is a CA waveform if and only if

∀m ∈ ZN ,

∣∣∣∣∣N−1∑

n=0

Au(m,n)

∣∣∣∣∣ = 1

Theorem 6.5 is clear by matrix multiplication. It is useful because it says that a signal’s ambiguity

function data, which can be recorded in radar, allows us to compute the signal.

Theorem 6.5. Let u : ZN → Cd and let Au be its ambiguity function matrix defined in Definition

1.2. Define the N ×N matrix U = (Ui,j), where Ui,j =< u(i + j), u(j) >. Then U = Au DN .

Proposition 6.6. Let u = {u(k)}N−1k=0 ⊆ C and let Au be its ambiguity function matrix. Then

u is a ZAC waveform if and only if Au has an eigenvalue of 1 with corresponding eigenvector

[1 0 0 . . . 0].

Remark 6.7. a. Let u : ZN → C and consider the matrix U of Theorem 6.5. Note that Uk,0 =

u(k)u(0). Hence, if we know the values of the ambiguity function, and, thus, the ambiguity function

matrix Au, then Theorem 6.5 allows us to retrieve the sequence u, which generates it, as long as19

u(0) 6= 0. In fact, if u(0) = 1 then u(k) = (AuDN )(k, 0).

b. Theorem 6.5 establishes a bijection between `2(ZN ) and a subset of the set of complex

N × N matrices CN×N . This allows us to convert a problem about sequences into a matrix

problem. In particular, we may define an equivalence relation on CN×N by saying that A ∈ CN×N

and B ∈ CN×N are equivalent if the entries of A may be obtained from B by applying a finite

succession of the operations derived in Theorem 2.1 a–e. Therefore, equivalent ambiguity function

matrices correspond to equivalent sequences. Thus, finding the number of nonequivalent ambiguity

function matrices will determine the number of nonequivalent sequences. In fact, by utilizing the

restrictions imposed by Propositions 6.1, 6.2, 6.3, 6.4, and 6.6, this method is successful at finding

all nonequivalent sequences for small values of n.

c. Theorem 6.5 may also be used to construct new equivalence classes of CAZAC waveforms by

choosing an appropriate matrix to serve as the ambiguity function matrix. Note that an arbitrary

matrix may not be the ambigutiy function matrix of some CAZAC waveform, or even any complex

sequence. Propositions 6.2 and 6.3 place restrictions on the structure of a matrix in order for it to

be an ambiguity function matrix. Propositions 6.1, 6.4, and 6.6 provide analogous restrictions in

the CAZAC and ZAC cases.

Example 6.8. In practice, given a sequence u = {u(k)}N−1k=0 , a fixed n ∈ Z, and the values of its

ambiguity function Au(m,n) for all m ∈ ZN , one may wish to find n, which can be thought of as

an observable quantity, e.g., related to a Doppler shift. If u is a Wiener CAZAC waveform then

Corollary 3.4 allows us to calculate n mod N . If we can use several different waveforms, then the

following observation allows for an alternative method for calculating n mod N .

Let n ∈ Z and let {{uj(k)}Nj−1k=0 }M

j=1 be a sequence of Wiener CAZAC waveforms uj : ZNj →C with mutually relatively prime lengths Nj and with corresponding ambiguity function values

{{Auj (m,n)}Nj−1m=0 }M

j=1. Using Corollary 3.4 for each j = 1, . . . , M we can calculate n mod Nj .

Consequently, we can use the Chinese Remainder Theorem to find n mod∏M

j=1 Nj .

Let N =∏M

j=1 paj

j be the prime decomposition of N . Given n ∈ Z and a Wiener CAZAC

waveform {u(k)}N−1k=0 , Corollary 3.4 allows us to calculate n mod N using N2 =

∏Mj=1 p

2aj

j inner

product operations. Alternatively, the Chinese Remainder Theorem method allows us to calculate

n mod N using only∑M

j=1 p2aj

j inner product operations.

Theorem 6.9. Let x : ZK → C be a ZAC waveform. Suppose 2J + 1 < K = 2L + 1 and let

Z(J) = {−J, . . . , J}. Consider the set BL of all waveforms z : ZK → C with the property that

supp (z) ⊆ Z(J). The mean-square-error (MSE), ||z − x||, for z ∈ BL has a minimum value of20

(2(L− J))1/2, and this minimum value is attained by y = x ∗ dJ ∈ BL, where dJ = 1Z(J), i.e.,

∀z ∈ BL, (2(L− J))1/2 = ||y − x|| ≤ ||z − x||.

Proof. Let y = x ∗ dJ . Then supp (y) ⊆ Z(J). We compute

||y − x||2 = ||y − x||2 =L∑

k=−L

|y(k)− x(k)|2

=∑

J<|k|≤L

|x(k)|2 +J∑

k=−J

(y(k)− x(k))(y(k)− x(k))

=∑

J<|k|≤L

|x(k)|2.

This clearly minimizes ||y − x||. Since x is a ZAC waveform, x is a CA waveform. Hence, we have

||y − x|| = ∑

J<|k|≤L

|x(k)|2

1/2

= (2(L− J))1/2.

¤

Acknowledgements

The authors wish to thank Drs. Douglas Cochran and Bahman Saffari and the following scientists

at the Norbert Wiener Center: Drs. Michael Dellomo, Andrew Kebo, Ioannis Konstantinidis,

and Jeffrey Sieracki. We are also grateful for support from the University of Maryland VIGRE

Grant DMS0240049 for the second named author, as well as from ONR Grant N00014-06-1-0002,

AFOSR(MURI) Grant FA9550-05-1-0443, and DARPA funding under NRL Grant N00173-06-1-

G006.

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Figure 1. Ambiguity function modulus |Au(•, n)|, as a function of m, of a length42 Wiener CAZAC waveform u evaluated at n = 1,2,3,4, where u(k) = ωk2

forω = e2πi/84.

24

Figure 2. Ambiguity function modulus |Au(•, n)|, as a function of m, of a length33 Wiener CAZAC waveform u evaluated at n=1,2,3,4, where u(k) = ωk2

for ω =e2πi/33.

25

Figure 3. Ambiguity function domain of a length 75 Wiener CAZAC waveformu, where |Au(m,n)| = 1 for a dot (m,n) and Au(m, n) = 0 otherwise, and whereu(k) = ωk2

for ω = e2πi/75.

Figure 4. Ambiguity function domain of a length 128 Wiener CAZAC waveformu, where |Au(m,n)| = 1 for a dot (m,n) and Au(m, n) = 0 otherwise, and whereu(k) = ωk2

for ω = e2πi/256.

26

Figure 5. Ambiguity function domain of a length 100 Wiener CAZAC waveformu, defined in Theorem 3.3 and for j =2,3,4,9,25,98.

27

Figure 6. Ambiguity function modulus |Au(•, n)|, as a function of m, of length3× 52 , 9× 32, 5× 32, and 5× 42 Milewski CAZAC waveforms evaluated at n =1,5, 5, and 7, respectively.

28

Figure 7. Ambiguity function modulus |Au(•, n)|, as a function of m, of a length4× 32 Milewski CAZAC waveform evaluated at n = 1, 2, 3, 4.

29

Figure 8. Ambiguity function modulus |Au(•, n)|, as a function of m, of a length5×32 Milewski CAZAC waveform, generated by a length 5 Wiener CAZAC waveformand evaluated at n = 1, 2, 3, 4, 5, 6.

30